the vector

ABSTRACT

THE INVENTION HEREIN DESCRIBED WAS MADE IN THE COURSE OF OR UNDER A CONTRACT OR SUBCONTRACT THEREUNDER, WITH THE DEPARTMENT OF THE NAVY. THE PRESENT INVENTIONS RELATES TO A METHOD FOR FINDING OPTIMUM INTEGER SOLUTIONS TO LINEAR PROGRAMS. IN MANY LINEAR PROGRAMS, IT IS NECESSARY TO OBTAIN INTEGER VALUES FOR CERTAIN VARIABLES. STARTING WITH A SOLUTION OF THE LINEAR PROGRAM OBTAINED, FOR EXAMPLE, BY THE SIMPLEX METHOD, THE PRESENT METHOD DEVELOPS BOUNDS WHICH ARE UTILIZED TO FORCE CERTAIN VARIABLES TO BE INTEGERS WITH A MINIMUM OF MACHINE CALCULATIONS.

DEFENSIVE PUBLICATION UNITED STATES PATENT OFFICE Published at therequest of the applicant or owner in accordance with the Notice of Dec.16, 1969, 869 0.G. 887. The abstracts of Defensive Publicationapplications are identified by distinctly numbered series and arearranged chronologically. The heading of each abstract indicates thenumber of pages of specification, including claims and sheets ofdrawings contained in the application as originally filed. The files ofthese applications are available to the public for inspection andreproduction may be purchased for 30 cents a sheet.

Defensive Publication applications have not been examined as to themerits of alleged invention. The Patent Ofilce makes no assertion as tothe novelty oi! the disclosed subject matter.

PUBLISHED JANUARY 7, 1975 T930.001 METHOD FOR FINDING OPTIMUM INTEGER-VALUED SDLUTION 0F LINEAR PROGRAMS Ralph E. Gomory, Chappaqua, and EllisL. Johnson, Yorktown Heights, N.Y., assignors to International BusinessMachines Corporation, Armonk, N.Y. Continuation of application Ser. No.158,462, June 30, 1971. This application May 29, 1973, Ser. No. 364,396Int. Cl. G06f 15/34 US. Cl. 444-1 7 Sheets Drawing. 108 PagesSpecification srspi HiGH LEVEL FLOWCHART STEP 2 sum AA UIISOLVEDLIIIEAII PROGRAM.

15 mi LIIIEAR PROCRAH IHFEASIBLE I no as ours lHE oPnxuu UNEARPRDGRAIUIINB SOLUIIOM sAIIsrI HIE INTEGER PRIIcRAIu l o 1E5 STEP 3 (TESTTHIS LIAEAR PROGRAM l5 lHIS sIILuIIoII lNE EESI M 1 amass A sum LINEARmum souIIwII rouun PRDGRAN. so on? moan II As HIE nsr nuns nus LHIEARPROGRAM TERIUWE IF HERE ARE H0 Hm LINEAR mums. seLvinoR UNSULVED- WEIIAIE lNE TERHIHAIE. oIIImIsz, EllNER sour SGME UMSCLVED LINEAR HIS IIIE PIwcRAII [STEP 2). 0R UEYEEWINE A acurw l!) sou: SOLVED LmSOLUIIUK-HFWE EAR PRocRAII SiEP 4), or mi $0M SOLVEll LINiAFl PROGRAMElms) mm on on MORE UNSOLVED LIIIEAR PROGRAMS (STEP 51.

FOR sous sen Ito UI'EI'EAYR PIRZGRIYlM, on IT BE DI; {MIND IIA vm mumsaunou HAS OEZJUZHVE $TEP4 N0 VALUE AI LEIlSlqAS LARGE As lHE F5PRO-GRAN BEST Isms soLIInou IBIIuIInI 5a m? ADJOIN ous on HDRERESTRIEIIONS l0 sou: l STEP 5 SOLVED LINEAR PRocnAII l0 cw: our on MOREIlium" on UNSOLVED LINEAR PROGRAM The Invention herein described wasmade in the course of or under a contract or subcontract thereunder,with the Department of the Navy.

The present invention relates to a method for finding optimum integersolutions to linear programs. In many linear programs, it is necessaryto obtain integer values for certain variables. Starting with a solutionof the linear program obtained, for example, by the Simplex Method, thepresent method develops bounds which are utilized to force certainvariables to be integers with a minimum of machine calculations.

jam. 7, 1975 R. E. GOMORY EY'AL 'mwmm METHOD FOR FINDING OPTIMUMINTEGER-VALUED SOLUTION OF LINEAR PROGRAMS Original Filed June 30, 1971'7 Sheets-Sheet l FIG.1

INVENTORS RALPH E. GOMORY ELLIS L. JOHNSON ATTORNEY R E. GOMORY ETI'ALSTEP 1 MODEL FORMULATION SOLVE AN UNSOLVED LINEAR PROGRAM.

METHOD FOR I INDING OPTIMUER INTEGER-VALUED SOLUTION OF LINEAR PROGRAMSOriginal Filed June 30, 1971 7 Sheets-Sheet 2 IS THIS SOLUTION THE BESTINTEGER SOLUTION FOUND SO FAR? YES E ISTHE LINEAR PROGRAM INFEASIBLE ,NOYES OOEs IIIE OPIIMIIM LINEAR PROGRAMMING sOLIIIION SATISFY IRE INIEGERPROGRAM? NO YES STEP 3 I (TEST THIS LINEAR PROGRAM THESOLUTIONI BECOMESA sOLvEO LINEAR PROGRAM.

II'ES REGORO IIAs IIIE REsI DELETE IRIs LINEAR PROGRAM EBLLWAIE II INEREARE NO MORE LINEAR PROGRAMs, SOLVED OR uNsOLvEO, NE IIAIIE IRETERMINATE. OINER IIIsE, EIINER SOLVE sOME IINsOLvEO LINEAR BE INTEGERPROGRAM (STEP 2), OR IIEIERIIINE A BOUND IO sOME sOLvEO LIN- ROLUTION- IIFOIIE EAR PROGRAM I sIEP 4), OR MAIIE sOME sOLvEO LINEAR PROGRAMEXISTS) INIO ONE OR MORE UNSOLVED LINEAR PROGRAMS IsIEP 5 I.

FOR SOME sOLvEO LINEAR PROGRAM, GAN II BE DETERMINED INAI EvERI INIEGERSOLUTION HA5 OBJECTIVE STEP 4 NO MORE AI LEAsI AS LARGE As IIIE REsIINIEGER sOLIIIION (BOUND) SOFAR? AOIOIN ONE OR MORE REsIRIGIIONs IO sOMESTEP 5 sOLvEO LINEAR PROGRAM IO GIvE ONE OR MORE IRIIANGR OR UNSOLVEDLINEAR PROGRAM.

CUT)

E. GOMORY ET AL 19301101 METHOD FOR FINDING OPTIMUM INTEGER-VALUEDSOLUTION OF LINEAR PROGRAMS 1971 7 Sheets-Sheet. 1'

Original Filed June 50,

R m m N m m m N F O q fl x u\ x W x AV m U) Av Q Av I I A AV mm I I !o.Ill AW Ill v I ll um II O Ill {I a II. A? x QM} 19E; Cuff: M Li .TEARR.7, 197$ Original Filed JuI R. E. GGMORY ETA!- METHOD FOR FINDING OPTIMUMINTEGER-VALUED SOLUTION OF LINEAR PROGRAMS TQSQOOA '7 Sheets-Sheet 4CHOOSE THE LINEAR PROGRAM HAVING THE SMALLEST VALUE Z OF THE OBJECTIVEFUNCTION AND WHICH HAS NOT YET BEEN BRANCHED ON. IDENTIFY THIS LINEARPROCRAMAS THE BRANOHING LINEAR PROGRAM.

IS THE OBJECTIVE VALUE Z SMALLER THAN THE BEST INTEGER OBJECTIVE so FAR4 NO YES 6 1553M Is ANY BASIC VARIABLE AN INTEGER THE REsT INTEGERSOLUTION VARIABLE wRosE BASIC VALUE Is NoT I5 OPTIMUM. IF NONE HAS BEENAN INTEGER FOUND, TREN TRERE Is N0 INTEGER SOLUTION. NO YES RECORD THISINTEGER SOLUTION AS THE BEST FOUND SO FAR.

I 8 THE BRANCHING LINEAR PROGRAM HAS Now BEEN BRANCHED 0N.

DOES THERE RENAIN ANY LINEAR PRIIGRAII WHICH Is FEASIBLE AND HASNOT REENBRANCHED 0N INO YES Jan. 7, 1975 R E. G OMORY ETAL METHOD FOR FINDINGOPTIMUM INTEGEIFVALUED Original Filed June 30,

SOLUTION OF LINEAR PROGRAMS 197] '7 Sheets-Sheet 5 FOR EACH INTEGERBASIC VARIABLE AT A FRAGTIONAL VALUE, GOMPUTE THE BOUND. LET X BE THEBASIQVARIABLE GIVING THE LARGEST BOUND, WHERE XI i bi IS NOT AN INTEGER.LET Z5 ZLP THE BOUND FROM X I IO IS 2 SMALLER THAN THE BEST INTEGEROBJECTIVE vALIIE YES soLvE THE Two LINEAR PROGRAMS GIvEN BY EIxIIIG x ATL biJJ INTEGER ITELGNB AN II AT IBfl, THE INTEGER ABOVE b IE EITHERLINEAR PROGRAM IS INFEASIBLE, DELETE IT IHIIEIIIATLI. OTHERWISE, REGoRIIIL AS THE VARIABLE EIxEII To FORM THELINEAR PROGRAMS AND REGoRII x ASBEING INCREASINGFOR THE LIIIEAR PROGRAM FORMED BY x =rI rI ANDDECREASING FOR x LET X BE THE VARIABLE FIXED TO FORM THE BRANGHINGLINEAR PROGRAM SAY X} v. SOLVE ANEW LINEAR PROGRAM BY FIXING X 'IORV 1DEPENDING ON WHETHER X IS IN' GREASING OR DECREASING. IF THIS IIEWLINEAR PROGRAM IS INFEASIBLE, DELETE IT IMMEDIATELY.

FIG.4B

flan. "FE, 1%75 R EGOMORY ETAL 'WMEWM METHOD FOR FINDING OPTIMUMINTEGEIPVALUED SOLUTION OF LINEAR PROGRAMS Original Filed Juna 30, 19717 Sheets-Sheet 7 APL#3O l5 N YEs APL #31 1r 1r e x (f f J2 APL #32 77':7T 6 X (f -f #34 IS 6 Tr 0R c If? NO YES ,14 #55 LET INEW BE THE IGIVING THE LET THE FOLLOWING MINIMUM IN THE EXPRESSION FOR DEFINE k ,k,j ,AND Tu #33 E (k {E (i):E (i) f 46 #35 I 'IIEIII A5 E III I=:"{EIII=E I II I NO YES #48 E I I='?"{E III=E II f I I} #48 E I I="{"{EIII=E III I I I} #51 E =(E ,f (J #58 PI=(P ,c (j W) 19 I f NE THE BOUNDIs LET LE NUMBER OF ENTRIES IN EI #53 {p (k )+1r (f E (k p (k )+7r"(E (k)f #59 RE-INDEX EI(AND PI) so THAT /20 #40 E1 IS INCREASING THEINEQUALITY Is 2 W mpg-Z 1r'IL.. #46 HI -I j JN-J J #49 ZmiH{P1I I+ II 1I-E I I),

21 M) (EIXj )-f lj))}X 41 {I =(f ,F(f (j w) EI(2,...,LE-1)) 4 c =IC ,ICI .,I-E I2,. ,L -III ZBOUND

